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Theorem eluniab 3639
Description: Membership in union of a class abstraction. (Contributed by NM, 11-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
eluniab (𝐴 {𝑥𝜑} ↔ ∃𝑥(𝐴𝑥𝜑))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem eluniab
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eluni 3630 . 2 (𝐴 {𝑥𝜑} ↔ ∃𝑦(𝐴𝑦𝑦 ∈ {𝑥𝜑}))
2 nfv 1462 . . . 4 𝑥 𝐴𝑦
3 nfsab1 2073 . . . 4 𝑥 𝑦 ∈ {𝑥𝜑}
42, 3nfan 1498 . . 3 𝑥(𝐴𝑦𝑦 ∈ {𝑥𝜑})
5 nfv 1462 . . 3 𝑦(𝐴𝑥𝜑)
6 eleq2 2146 . . . 4 (𝑦 = 𝑥 → (𝐴𝑦𝐴𝑥))
7 eleq1 2145 . . . . 5 (𝑦 = 𝑥 → (𝑦 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑥𝜑}))
8 abid 2071 . . . . 5 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
97, 8syl6bb 194 . . . 4 (𝑦 = 𝑥 → (𝑦 ∈ {𝑥𝜑} ↔ 𝜑))
106, 9anbi12d 457 . . 3 (𝑦 = 𝑥 → ((𝐴𝑦𝑦 ∈ {𝑥𝜑}) ↔ (𝐴𝑥𝜑)))
114, 5, 10cbvex 1681 . 2 (∃𝑦(𝐴𝑦𝑦 ∈ {𝑥𝜑}) ↔ ∃𝑥(𝐴𝑥𝜑))
121, 11bitri 182 1 (𝐴 {𝑥𝜑} ↔ ∃𝑥(𝐴𝑥𝜑))
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103  wex 1422  wcel 1434  {cab 2069   cuni 3627
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-uni 3628
This theorem is referenced by:  elunirab  3640  dfiun2g  3736  inuni  3956  snnex  4234  elfv  5250  unielxp  5878  tfrlem9  6015  tfr0dm  6018
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